An inequality on the matrix spectral norm specifying Hadamard matrices
Eugene C. Johnsen · Linear and Multilinear Algebra · 1977
Recently, C. A. Akemann raised the question of how much the spectral norm of a real matrix of order n can increase when its entries are replaced by values having absolute values no larger than those of the corresponding original entries. If Ais the original matrixà is the matrix with entries replaced, and are the values of the spectral norm on Aand Ã, respectively, then it is easy to show that . In this paper we place this question in a slightly more general setting and for complex matrices we characterize the case of equality in the resulting inequality. When the original matrix A≠ 0 and the matrix of replaced entries à are real, the case of equality holds if and only if A is a real scalar multiple of a Hadamard matrix and à is of rank 1 with the absolute values of the entries of A uniformly related to those of A. From this we obtain a characterization of Hadamard matrices within the class of all real matrices in terms of an inequality on the spectral norm.